A tap fills a tank in 12 hours and the other empties it in 24 hours. If both are opened simultaneously, then the tank will be filled in :

Options

A. 42 hours
B. 20 hours
C. 24 hours
D. 22 hours

Answer

To solve the problem of how long it takes to fill a tank when one tap fills it in 12 hours and another tap empties it in 24 hours, we can follow these steps:

### Step-by-Step Solution:

1. **Determine the Filling Rate of the First Tap:**
- The first tap fills the tank in 12 hours.
- Therefore, in 1 hour, it fills \( \frac{1}{12} \) of the tank.

2. **Determine the Emptying Rate of the Second Tap:**
- The second tap empties the tank in 24 hours.
- Therefore, in 1 hour, it empties \( \frac{1}{24} \) of the tank.

3. **Calculate the Net Rate of Filling When Both Taps are Open:**
- When both taps are open, the net amount of the tank filled in 1 hour is:
\[ \text{Net Rate} = \text{Filling Rate} - \text{Emptying Rate} = \frac{1}{12} - \frac{1}{24} \]
- To perform this subtraction, we need a common denominator. The least common multiple of 12 and 24 is 24.
- Convert \( \frac{1}{12} \) to have a denominator of 24:
\[ \frac{1}{12} = \frac{2}{24} \]
- Now, subtract:
\[ \frac{2}{24} - \frac{1}{24} = \frac{1}{24} \]

4. **Determine the Time to Fill the Tank:**
- Since the net rate of filling the tank is \( \frac{1}{24} \) of the tank per hour, it will take:
\[ \text{Time} = \frac{1 \text{ tank}}{\frac{1}{24} \text{ tank/hour}} = 24 \text{ hours} \]

### Final Answer:
The tank will be filled in **24 hours**.

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